{
  "case_id": "case_assignment",
  "problem_type": "assignment",
  "natural_language": "Assign 4 workers to 4 tasks. Processing times: Worker 1 takes 9, 2, 7, 8 hours; Worker 2 takes 6, 4, 3, 7; Worker 3 takes 5, 8, 4, 6; Worker 4 takes 7, 5, 6, 4. Each worker handles exactly one task and each task is assigned to exactly one worker. Minimize total processing time.",
  "binding": {
    "problem_hint": "assignment",
    "entities": {
      "workers": [
        {
          "id": "W1",
          "label": null,
          "attributes": {}
        },
        {
          "id": "W2",
          "label": null,
          "attributes": {}
        },
        {
          "id": "W3",
          "label": null,
          "attributes": {}
        },
        {
          "id": "W4",
          "label": null,
          "attributes": {}
        }
      ],
      "tasks": [
        {
          "id": "T1",
          "label": null,
          "attributes": {}
        },
        {
          "id": "T2",
          "label": null,
          "attributes": {}
        },
        {
          "id": "T3",
          "label": null,
          "attributes": {}
        },
        {
          "id": "T4",
          "label": null,
          "attributes": {}
        }
      ]
    },
    "parameters": {
      "processing_time": {
        "name": "processing_time",
        "value": [
          [
            9.0,
            2.0,
            7.0,
            8.0
          ],
          [
            6.0,
            4.0,
            3.0,
            7.0
          ],
          [
            5.0,
            8.0,
            4.0,
            6.0
          ],
          [
            7.0,
            5.0,
            6.0,
            4.0
          ]
        ],
        "unit": null,
        "source_span": null
      }
    },
    "scalars": {},
    "binding_confidence": 0.9
  },
  "formulation": {
    "problem_type": "assignment",
    "sets": {
      "I": "workers",
      "J": "tasks"
    },
    "parameters": [
      "processing_time"
    ],
    "variables": [
      {
        "name": "x",
        "indices": [
          "i",
          "j"
        ],
        "domain": "binary",
        "non_negative": true,
        "description": "1 if worker i assigned to task j"
      }
    ],
    "objective": {
      "sense": "minimize",
      "expression": "sum(processing_time[i,j] * x[i,j] for i in I for j in J)",
      "description": "Minimize total processing time"
    },
    "constraints": [
      {
        "name": "one_task_per_worker",
        "expression": "sum(x[i,j] for j in J)",
        "sense": "==",
        "rhs_expression": "1",
        "description": "Each worker gets exactly one task"
      },
      {
        "name": "one_worker_per_task",
        "expression": "sum(x[i,j] for i in I)",
        "sense": "==",
        "rhs_expression": "1",
        "description": "Each task assigned to exactly one worker"
      }
    ],
    "notes": []
  },
  "ground_truth_objective": 14.0
}